1.06g Equations with exponentials: solve a^x = b

483 questions

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CAIE P3 2016 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(4 ^ { 3 x - 1 } = 3 \left( 5 ^ { x } \right)\), giving your answer correct to 3 decimal places.
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { 2 - x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
  2. Calculate the exact \(x\)-coordinate of the point of intersection of this line with the line with equation \(y = 2 x\), simplifying your answer.
CAIE P3 2017 June Q3
4 marks Moderate -0.8
3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.
CAIE P3 2019 June Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(9 ^ { x } = 3 ^ { x } + 12\). Give your answer correct to 2 decimal places.
CAIE P3 2019 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 3 - 2 x } = 4 \left( 7 ^ { x } \right)\), giving your answer correct to 3 decimal places.
CAIE P3 2016 March Q1
3 marks Standard +0.3
1 Solve the equation \(\ln \left( x ^ { 2 } + 4 \right) = 2 \ln x + \ln 4\), giving your answer in an exact form.
CAIE P3 2017 March Q1
3 marks Moderate -0.5
1 Solve the equation \(\ln \left( 1 + 2 ^ { x } \right) = 2\), giving your answer correct to 3 decimal places.
CAIE P3 2006 November Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) satisfying the inequality \(\left| 3 ^ { x } - 8 \right| < 0.5\), giving 3 significant figures in your answer.
CAIE P3 2008 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 2 ) = 2 + \ln x$$ giving your answer correct to 3 decimal places.
CAIE P3 2009 November Q2
4 marks Moderate -0.8
2 Solve the equation \(3 ^ { x + 2 } = 3 ^ { x } + 3 ^ { 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2009 November Q1
4 marks Moderate -0.5
1 Solve the equation $$\ln ( 5 - x ) = \ln 5 - \ln x$$ giving your answers correct to 3 significant figures.
CAIE P3 2010 November Q2
4 marks Standard +0.3
2 Solve the equation $$\ln \left( 1 + x ^ { 2 } \right) = 1 + 2 \ln x$$ giving your answer correct to 3 significant figures.
CAIE P3 2011 November Q1
4 marks Moderate -0.8
1 Using the substitution \(u = \mathrm { e } ^ { x }\), or otherwise, solve the equation $$\mathrm { e } ^ { x } = 1 + 6 \mathrm { e } ^ { - x }$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 November Q2
4 marks Standard +0.3
2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 November Q1
3 marks Moderate -0.5
1 Solve the equation $$\ln ( x + 5 ) = 1 + \ln x$$ giving your answer in terms of e.
CAIE P3 2013 November Q2
4 marks Standard +0.3
2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.
CAIE P3 2014 November Q1
3 marks Moderate -0.8
1 Use logarithms to solve the equation \(\mathrm { e } ^ { x } = 3 ^ { x - 2 }\), giving your answer correct to 3 decimal places.
CAIE P3 2015 November Q2
5 marks Moderate -0.3
2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.
CAIE P3 2016 November Q1
3 marks Moderate -0.5
1 Solve the equation \(\frac { 3 ^ { x } + 2 } { 3 ^ { x } - 2 } = 8\), giving your answer correct to 3 decimal places.
CAIE P3 2017 November Q2
5 marks Standard +0.3
2 Showing all necessary working, solve the equation \(2 \log _ { 2 } x = 3 + \log _ { 2 } ( x + 1 )\), giving your answer correct to 3 significant figures.
CAIE P3 2018 November Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } = 4\), giving your answer correct to 2 decimal places.
CAIE P3 2019 November Q1
3 marks Standard +0.3
1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).
CAIE P3 2019 November Q1
3 marks Moderate -0.3
1 Solve the equation \(5 \ln \left( 4 - 3 ^ { x } \right) = 6\). Show all necessary working and give the answer correct to 3 decimal places.
CAIE P3 2019 November Q3
4 marks Standard +0.3
3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.
CAIE P3 2019 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
  2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
  3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.